In a statement on Wednesday afternoon, House Speaker John Boehner confirmed that House Republicans won't block passage of the deal, thereby clearing the way for the federal government to reopen and for the U.S. to raise its debt ceiling.

The bill will also include an income-verification system for individuals and families receiving subsidies through the Affordable Care Act. In short, the Secretary of Human Health Services will submit a report no later than Jan. 1 detailing the verification process. The department's inspector general will submit to Congress no later than July a report on the effectiveness of those procedures.

And in what was a key priority for the White House, the Treasury Department will still be able to use "extraordinary measures" to work around the debt ceiling in the case that it is not raised by Feb. 7.

"The compromise we reached will provide our economy with the stability it desperately needs," Reid said on the Senate floor.

McConnell spoke in a more somber tone on the floor after Reid, but he also hailed the fact that the deal keeps in place the across-the-board spending cuts of the sequester.

"For today, the relief we hope for is to reopen the government, avoid default, and protect the historic cuts we achieved under the [Budget Control Act]," McConnell said of the bill that came out of the 2011 debt-ceiling crisis.

"This is far less than many of us had hoped for. But it’s far better than what some had sought. Now it’s time for Republicans to unite behind our other crucial goals."

Boehner said in his statement that the House had done everything it could to fight Obamacare.

"That fight will continue. But blocking the bipartisan agreement reached today by the members of the Senate will not be a tactic for us," Boehner said in the statement.

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Harry Reid

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Sen. Ted Cruz (R-Texas) told reporters shortly after the deal was announced that he does not intend to delay passage of the Senate legislation. He said that it's a time for Republicans to announce "where they stand" on the Affordable Care Act, but that he never had any intention of delaying what has become inevitable.

"There's nothing to be gained from delaying this vote one day or two days," Cruz said.

A Senate Democratic leadership aide said that the Senate will act first on the legislation — likely sometime late this afternoon or early this evening. Then the House will act afterward.

The White House applauded the Senate deal, and Press Secretary Jay Carney said that Obama commended both Reid and McConnell for working to forge an agreement.

But Carney didn't say whether the White House expected a vote in both chambers Wednesday.

" We are not putting odds on anything," Carney said. He added that Obama expected Congress to "act swiftly."